Algebraic Geometry Seminar
Spring 2016 — Tuesdays 3:30  4:30 PM, location JFB 102
Date  Speaker  Title — click for abstract (if available) 
January 19 
Junecue Suh University of California, Santa Cruz 
Vanishing theorems for mixed Hodge modules
We discuss two vanishing theorems for mixed Hodge modules, one of EsnaultViehweg type and the other of KawamataViehweg type.

January 26 2:50  3:50PM JFB 102 
Nicolas Addington University of Oregon 
Lehn's hyperkahler 8fold associated to a cubic 4fold
Compact hyperkahler varieties are higherdimensional analogues of K3 surfaces. Very few examples are known: two infinite series and two sporadic examples.
At the same time we cannot prove that there are not many more examples. In 2013, Lehn et al. constructed a family of hyperkahler 8folds Z from the variety
of twisted cubic curves on a cubic 4fold. I will discuss some joint work with Lehn from 2014 and some new work, which together clarify Z's place in the moduli
space of hyperkahlers. In particular it is deformationequivalent, but typically not isomorphic or even birational, to the Hilbert scheme of 4 points on a K3 surface.
A crucial role is played by Kuznetsov's K3 subcategory of the derived category of the cubic.

February 2 2:50  3:50PM JFB 102 
Burt Totaro University of California, Los Angelos 
Hypersurfaces that are not stably rational
We show that a large class of complex hypersurfaces
in all dimensions are not stably rational.
(A variety is stably rational if its product with some projective space
is birational to projective space.) Namely, for all d at least about 2n/3,
a very general hypersurface of degree d in P^{n+1}
is not stably rational. The result covers all the degrees in which Kollar
proved that a very general hypersurface is not rational, and a bit more.
For example, very general quartic 4folds are not stably rational,
whereas it was not even known whether these varieties are rational.

February 9 2:50  3:50PM JFB 102 
Xudong Zheng University of Illinois at Chicago 
The Hilbert scheme of points on singular surfaces
The Hilbert scheme of points on a quasiprojective variety parameterizes its zerodimensional subschemes. These Hilbert schemes are smooth and irreducible for smooth surfaces but will eventually become reducible for sufficiently singular surfaces. In this talk, I provide the first class of examples of singular surfaces whose Hilbert schemes of points are irreducible, namely surfaces with at worst cyclic quotient rational double points. I will also describe some consequent geometric properties of these irreducible Hilbert schemes.

February 16 2:50  3:50PM JFB 102 
Hamid Hassanzadeh Federal University of Rio de Janeiro 
Degree of the inverse of birational maps via the syzygies of the base loci
This is a classical fact that if a plane Cremona map F:P^2>P^2 is given by Polynomials of degree d then so does its inverse.
A generalization of this fact for birational maps from P^n to P^n is due to Gabber who showed in 80's that the inverse map has
degree at most d^{n1}. However if one considers an arbitrary projective variety X of P^n, then bounding the degree of the inverse map becomes a subtle challenge.
This is the leitmotiv of this talk which answers a question by Jeremy Blanc.
We start from Plane Cremona maps to show how the Betti table of the base ideal constrains birationality. Moving from Plane case to higher dimensions, we encounter the
role of higher polynomial equations of the base ideal, consequently the structure of the Rees algebra appears. A new Criterion for Birationality is presented which provides
information on the inverse map for any birational map F:X>Y of projective varieties.

February 23 2:50  3:50PM LCB 219 
John Lesieutre University of Illinois at Chicago 
Dynamical MordellLang and automorphisms of higherdimensional varieties
The dynamical MordellLang conjecture predicts that if f is
an endomorphism of a complex variety X, with p a point of X and
V a subvariety, then the set of n for which f^n(p) lands in V is
a union of a finite set and finitely many arithmetic progressions. When
f is \'etale, this is a result of BellGhiocaTucker. I'll discuss
an extension of this result to the setting in which p and V are
nonreduced closed subschemes of X, and show how this statement can be
applied to study dynamically interesting (e.g. positive entropy)
automorphisms of complex varieties in higher dimensions. This is joint
work with Daniel Litt.

March 1 
Anthony VárillyAlvarado Rice University 
Level structures, abelian varieties, and Brauer groups of K3 surfaces
In 1922, Mordell showed that the set of rational points on an elliptic curve over Q has the structure of a finitely generated abelian group.
Surprisingly, there are only 15 possibilities for the torsion part of this group; this is a spectacular result of Mazur from 1977.
In this talk I will discuss some higherdimensional generalizations of results along Mazur's theorem, including recent joint work with Dan Abramovich
on abelian varieties, and ongoing work for K3 surfaces, where I will argue that a suitable analogue of the group of rational points is the Brauer group.

March 8 
Brooke Ullery University of Utah 
Measures of irrationality for hypersurfaces of large degree
The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line.
There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties.
The intuition is that the higher these numbers, the further the variety is from being rational.
I will discuss some of these definitions, and present joint work with Lawrence Ein and Rob Lazarsfeld.
Our main result is that if X is an ndimensional hypersurface of degree d at least 5/2 n,
then any dominant rational map from X to P^n must have degree at least d1.

March 15 
Spring Break 

March 22 


March 29 
Jennifer Park University of Michigan 
Effective Chabauty for symmetric powers of curves
Faltings' theorem states that curves of genus g>1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud,
one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense.
In 1985, Coleman showed that Chabauty's method, which works when the MordellWeil rank of the Jacobian of the curve is smaller than g,
can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry
to show that we can also give an effective bound on the number of rational points outside of the special set of the dth symmetric power of X,
where X is a curve of genus g > d, when the MordellWeil rank of the Jacobian of the curve is at most gd.

April 5 
Jirui Guo Virginia Tech 
Quantum sheaf cohomology on Grassmannians
Quantum sheaf cohomology arises as the OPE ring of A/2 twisted theories with (0,2) supersymmetry. It is a deformation of the ordinary quantum cohomology. Its structure is well understood for toric varieties, but few results exsited for other spaces. In this talk, I will introduce recent progress on the study of quantum sheaf cohomology on Grassmannians with a deformed tangent bundle, especially an algebraic treatment of the classical limit, known as polymology, and how quantum corrections can be inferred.

April 12 
Piotr Achinger University of Warsaw 
Liftability of varieties in characteristic p
Many â€œpathologiesâ€?of varieties in characteristic p do not occur if
the variety in question lifts to characteristic zero, or at least if it lifts
modulo p^2. We will review classical and recent results on
liftable varieties, discuss liftability of Frobenius split varieties,
and present some new
(quite elementary) examples of nonliftable varieties. This is joint
work with Maciej Zdanowicz.

April 19 
Yuan Wang University of Utah 
Generic Vanishing and Classification of Irregular Surfaces in Positive Characteristics
The EnriquesKodaira classification of surfaces is one of the most celebrated accomplishments toward understanding algebraic varieties.
A detailed classification of surfaces of general type, however, seems to be very difficult. Several cases of this kind are known in
characteristic 0. In my talk I would like to present a classification result for surfaces of general type in positive characteristics,
and in particular we work on surfaces with Euler characteristic 1 and Albanese dimension 4. This project is inspired by a paper of Hacon
and Pardini but contains a lot of new ideas, including the construction of a generic vanishing theorem for surfaces that lift to W_2(k),
the second Witt vectors.

April 26 
Yujong Tzeng University of Minnesota 
Algebraic cobordism group of divisors and counting singular
curves with tangency conditions
The algebraic cobordism theory constructed Morel and Levine
is a universal oriented BorelMoore homology theory for schemes.
Levine and Pandaripande developed an equivalent algebraic cobordism
theory and many results using degeneration methods in algebraic
geometry can be understood as invariants of this theory. In this talk
I will talk about the generalization of the algebraic cobordism theory
to bundles and divisors on varieties and discuss its application to
count singular curves with tangency conditions in varieties of any
dimension. This enumeration is motivated by Caporaso and Harris'
recursive formula for nodal curves with fixed tangency multiplicities
with a line on the complex projective plane.

This web page is maintained by Tiankai Liu, Mark Shoemaker, Nicola Tarasca, and YuChao Tu.